Andreas Strömbergsson (Uppsala): Maass waveforms and Galois representations

Abstract. The study of the spectrum of the Laplace operator on a hyperbolic surface is an important topic which has numerous applications for example in number theory and geometric equidistribution problems. The discrete eigenfunctions in this setting are called \textit{Maass waveforms} after Hans Maass who first studied them (1949). Of special interest are so-called \textit{arithmetic} hyperbolic surfaces; their spectra have several peculiar properties, one of which is the often occurring Maass waveforms of eigenvalue exactly $\frac 14$, related to certain \textit{Galois representations}. In some cases the existence of these Maass waveforms is still only conjectural (it would follow from a conjecture of Artin), but several important cases have been settled, in particular by results of Langlands and Tunnell (1980-81). The theorems of Langlands and Tunnell also play a small but crucial role in Wiles' proof (1995) of Fermat's Last Theorem.
In my talk I will attempt to explain some aspects of these matters in a way accessible to a general mathematical audience. I will also briefly describe some recent spectral computations by Andrew Booker (University of Michigan) and myself, where we were forced to construct several new explicit examples of Galois representations related to Maass waveforms.